A can complete one-third of a work in 10 days and B can do $\frac{3}{5}th$ of the same work in 24 days. They worked together for 10 days. The remaining work was completed by C alone in 15 days. In how many days can C alone do $\frac{2}{3}rd$ of the same work?

24

A = \(\frac{1}{3}\)W = 10 days = 30 days,

B = \(\frac{3}{5}\)W = 24 days = 40 days,

⇒ A + B worked for 10 days = (4 + 3) x 10 = 70 units.  ..(Efficiency × Days = Total work)

⇒ Remaining work = 120 - 70 = 50 units,

⇒ C completed 50 units in 15 days = \(\frac{50}{C}\) = 15 days,

⇒ Therefore, C = \(\frac{10}{3}\)  (Efficiency)

⇒ Time taken by C to complete \(\frac{2}{3}\) of total work = \(\frac{2}{3}\) x 120 = 80 units,

⇒ \(\frac{80}{10/3}\) = 24 days.    ..(\(\frac{Work}{Efficiency}\) = Time)